Ισοτροπική Ομάς
isotropy Group, stabilizer group
- Ένα είδος Ομάδων.
ΕτυμολογίαEdit
Η ονομασία "ισοτροπική" σχετίζεται ετυμολογικά με την λέξη "ισοτροπία".
ΕισαγωγήEdit
In mathematics, an action of a group is a way of interpreting the elements of the group as "acting" on some space in a way that preserves the structure of that space.
Common examples of spaces that groups act on are:
- sets,
- vector spaces, and
- topological spaces.
Actions of groups on vector spaces are called representations of the group.
Some groups can be interpreted as acting on spaces in a canonical way. For example, the symmetric group of a finite set consists of all bijective transformations of that set; thus, applying any element of the permutation group to an element of the set will produce another element of the set.
More generally, symmetry groups such as the homeomorphism group of a topological space or the general linear group of a vector space, as well as their subgroups, also admit canonical actions.
For other groups, an interpretation of the group in terms of an action may have to be specified, either because the group does not act canonically on any space or because the canonical action is not the action of interest.
For example, we can specify an action of the two-element cyclic group $ \mathrm{C}_2 = \{0,1\} $ on the finite set $ \{a,b,c\} $ by specifying that
- 0 (the identity element) sends $ a\mapsto a,b\mapsto b,c\mapsto c $, and that
- 1 sends $ a\mapsto b,b\mapsto a,c\mapsto c $.
This action is not canonical.
A common way of specifying non-canonical actions is to describe a homomorphism $ \varphi $ from a group G to the group of symmetries of a set X.
The action of an element $ g\in G $ on a point $ x\in X $ is assumed to be identical to the action of its image $ \varphi(g) \in \text{Sym}(X) $ on the point $ x $.
The homomorphism $ \varphi $ is also frequently called the "action" of G, since specifying $ \varphi $ is equivalent to specifying an action. Thus, if G is a group and X is a set, then an action of G on X may be formally defined as a group homomorphism $ \varphi $ from G to the symmetric group of X. The action assigns a permutation of X to each element of the group in such a way that:
- the identity element of G is assigned the identity transformation of X;
- any product gk of two elements of G is assigned the composition of the permutations assigned to g and k.
If X has additional structure, then $ \varphi $ is only called an action if for each $ g\in G $, the permutation $ \varphi(g) $ preserves the structure of X.
The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them.
In particular, groups can act on other groups, or even on themselves. Because of this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields.
Definition Edit
If G is a group and X is a set, then a (left) group action φ of G on X is a function
- $ \varphi\colon G \times X \to X\colon (g,x)\mapsto \varphi(g,x) $
that satisfies the following two axioms (where we denote φ(g, x) as g.x):^{[1]}
- Identity
- e.x = x for all x in X. (Here, e denotes the identity element of the group G.)
- Compatibility
- (gh).x = g.(h.x) for all g, h in G and all x in X. (Here, gh denotes the result of applying the group operation of G to the elements g and h.)
The group G is said to act on X (on the left). The set X is called a (left) G-set.
From these two axioms, it follows that for every g in G, the function which maps x in X to g.x is a bijective map from X to X (its inverse being the function which maps x to g^{−1}.x). Therefore, one may alternatively define a group action of G on X as a group homomorphism from G into the symmetric group Sym(X) of all bijections from X to X.^{[2]}
In complete analogy, one can define a right group action of G on X as an operation X × G → X mapping (x, g) to x.g and satisfying the two axioms:
- Identity
- x.e = x for all x in X.
- Compatibility
- x.(gh) = (x.g).h for all g, h in G and all x in X;
The difference between left and right actions is in the order in which a product like gh acts on x. For a left action h acts first and is followed by g, while for a right action g acts first and is followed by h. Because of the formula (gh)^{−1} = h^{−1}g^{−1} , one can construct a left action from a right action by composing with the inverse operation of the group. Also, a right action of a group G on X is the same thing as a left action of its opposite group G^{op} on X. It is thus sufficient to only consider left actions without any loss of generality.
Types of actions Edit
The action of G on X is called
- Transitive if X is non-empty and if for each pair x, y in X there exists a g in G such that g.x = y. For example, the action of the symmetric group of X is transitive, the action of the general linear group or the special linear group of a vector space V on V ∖ {0} is transitive, but the action of the orthogonal group of a Euclidean space E is not transitive on E ∖ {0} (it is transitive on the unit sphere of E, though).
- Faithful (or effective) if for every two distinct g, h in G there exists an x in X such that g.x ≠ h.x; or equivalently, if for each g ≠ e in G there exists an x in X such that g.x ≠ x. In other words, in a faithful group action, different elements of G induce different permutations of X. In algebraic terms, a group G acts faithfully on X if and only if the corresponding homomorphism to the symmetric group, G → Sym(X), has a trivial kernel. Thus, for a faithful action, G embeds into a permutation group on X; specifically, G is isomorphic to its image in Sym(X). If G does not act faithfully on X, one can easily modify the group to obtain a faithful action. If we define N = {g in G : g.x = x for all x in X}, then N is a normal subgroup of G; indeed, it is the kernel of the homomorphism G → Sym(X). The factor group G/N acts faithfully on X by setting (gN).x = g.x. The original action of G on X is faithful if and only if N = {e}.
- Free (or semiregular or fixed point free) if, given g, h in G, the existence of an x in X with g.x = h.x implies g = h. Equivalently: if g is a group element and there exists an x in X with g.x = x (that is, if g has at least one fixed point), then g is the identity. Note that a free action on a non-empty set is faithful.
- Regular (or simply transitive or sharply transitive) if it is both transitive and free; this is equivalent to saying that for every two x, y in X there exists precisely one g in G such that g.x = y. In this case, X is called a principal homogeneous space for G or a G-torsor. The action of any group G on itself by left multiplication is regular, and thus faithful as well. Every group can, therefore, be embedded in the symmetric group on its own elements, Sym(G). This result is known as Cayley's theorem.
- n-transitive if X has at least n elements and for all pairwise distinct x_{1}, ..., x_{n} and pairwise distinct y_{1}, ..., y_{n} there is a g in G such that g.x_{k} = y_{k} for k ≤ n. A 2-transitive action is also called doubly transitive, a 3-transitive action is also called triply transitive, and so on. Such actions define interesting classes of subgroups in the symmetric groups: 2-transitive groups and more generally multiply transitive groups. The action of the symmetric group on a set with n elements is always n-transitive; the action of the alternating group is n-2-transitive.
- Sharply n-transitive if there is exactly one such g.
- Primitive if it is transitive and preserves no non-trivial partition of X. See primitive permutation group for details.
- Locally free if G is a topological group, and there is a neighbourhood U of e in G such that the restriction of the action to U is free; that is, if g.x = x for some x and some g in U then g = e.
Furthermore, if $ G $ acts on a topological space $ X $, then the action is:
- Wandering if every point $ x \in X $ has a neighbourhood $ U $ such that $ \{g \in G : g \cdot U \cap U \neq \emptyset\} $ is finite.^{[3]} For example, the action of $ \mathbb Z^n $ on $ \mathbb R^n $ by translations is wandering. The action of the modular group on the Poincaré half-plane is also wandering.
- Properly discontinuous if $ X $ is a locally compact space and for every compact subset $ K \subset X $ the set $ \{g \in G: gK \cap K \neq \emptyset \} $ is finite. The wandering actions given above are also properly discontinuous. On the other hand, the action of $ \mathbb Z $ on $ \mathbb R^2 \setminus \{0\} $ by the linear map $ (x, y) \mapsto (2x, 1/2y) $ is wandering and free but not properly discontinuous.Πρότυπο:Sfn
- Proper if $ G $ is a topological group and the map from $ G \times X \rightarrow X \times X : (g,x) \mapsto (g \cdot x,x) $ is proper.^{[4]} If G is discrete then properness is equivalent to proper discontinuity for G-actions.
- Said to have discrete orbits if the orbit of each $ x \in X $ under the action of $ G $ is discrete in $ X $.^{[3]}
If X is a non-zero module over a ring R and the action of G is R-linear then it is said to be
- Irreducible if there is no nonzero proper invariant submodule.
Orbits and stabilizers Edit
Consider a group G acting on a set X. The orbit of an element x in X is the set of elements in X to which x can be moved by the elements of G. The orbit of x is denoted by G.x:
- $ G.x = \left\{ g.x \mid g \in G \right\}. $
The defining properties of a group guarantee that the set of orbits of (points x in) X under the action of G form a partition of X. The associated equivalence relation is defined by saying x ∼ y if and only if there exists a g in G with g.x = y. The orbits are then the equivalence classes under this relation; two elements x and y are equivalent if and only if their orbits are the same; i.e., G.x = G.y.
The group action is transitive if and only if it has only one orbit, i.e., if there exists x in X with G.x = X. This is the case if and only if G.x = X for all x in X.
The set of all orbits of X under the action of G is written as X/G (or, less frequently: G\X), and is called the quotient of the action. In geometric situations it may be called the orbit space, while in algebraic situations it may be called the space of coinvariants, and written X_{G}, by contrast with the invariants (fixed points), denoted X^{G}: the coinvariants are a quotient while the invariants are a subset. The coinvariant terminology and notation are used particularly in group cohomology and group homology, which use the same superscript/subscript convention.
Invariant subsetsEdit
If Y is a subset of X, we write GY for the set g.y : y ∈ Y and g ∈ G. We call the subset Y invariant under G if G.Y = Y (which is equivalent to G.Y ⊆ Y). In that case, G also operates on Y by restricting the action to Y. The subset Y is called fixed under G if g.y = y for all g in G and all y in Y. Every subset that is fixed under G is also invariant under G, but not vice versa.
Every orbit is an invariant subset of X on which G acts transitively. The action of G on X is transitive if and only if all elements are equivalent, meaning that there is only one orbit.
A G-invariant element of X is x ∈ X such that g.x = x for all |g ∈ G. The set of all such x is denoted X^{G} and called the G-invariants of X. When X is a G-module, X^{G} is the zeroth group cohomology group of G with coefficients in X, and the higher cohomology groups are the derived functors of the functor of G-invariants.
ΥποσημειώσειςEdit
- ↑ Eie & Chang (2010). [[[:Πρότυπο:Google books]] A Course on Abstract Algebra]. σελ. 144. Πρότυπο:Google books.
- ↑ This is done, e.g., by Smith (2008). [[[:Πρότυπο:Google books]] Introduction to abstract algebra]. σελ. 253. Πρότυπο:Google books.
- ↑ ^{3,0} ^{3,1} Thurston, William (1980), The geometry and topology of three-manifolds, Princeton lecture notes, σελ. 175, http://library.msri.org/books/gt3m/
- ↑ tom Dieck, Tammo (1987), Transformation groups, de Gruyter Studies in Mathematics, 8, Berlin: Walter de Gruyter & Co., σελ. 29, ISBN 978-3-11-009745-0, https://books.google.com/books?id=azcQhi6XeioC
Εσωτερική ΑρθρογραφίαEdit
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- mathworld.wolfram.com
- encyclopediaofmath.org
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